Parent Hamiltonian Construction of Generalized Calogero-Sutherland Models

TL;DR

This paper develops a method to construct parent Hamiltonians for certain trial states in one-dimensional quantum systems, using conformal field theory techniques, and applies it to models related to anyon physics.

Contribution

It introduces a reverse-engineering approach to derive continuum parent Hamiltonians for states described by rational conformal field theories, including Moore-Read and Read-Rezayi states.

Findings

Derived explicit continuum Hamiltonians for Moore-Read and Read-Rezayi states.

Connected the null-vector structure of primary fields to many-body annihilation operators.

Provided a framework for constructing Hamiltonians with specific zero modes.

Abstract

The Calogero-Sutherland model is a paradigmatic integrable system describing one-dimensional non-relativistic particles with inverse-square interactions. At interaction strength $\lambda=2$, the CSM exhibits a deep connection to anyon physics, featuring the Laughlin-Jastrow polynomial as its exact ground state. Motivated by this structure, we develop a general reverse-engineering construction of positive semi-definite continuum parent Hamiltonians for trial states admitting a rational conformal field theory description with central charge $c<1$. By leveraging the null-vector structure of the underlying primary fields and the associated Belavin-Polyakov-Zamolodchikov equations, we derive corresponding many-body annihilation operators. We then apply this construction explicitly to the Moore-Read and $k=3$ Read-Rezayi states - relating to Ising and Fibonacci anyons, respectively - obtaining continuum Hamiltonians for which these Jack-polynomial states are exact zero modes. We emphasize, however, that our construction does not by itself establish ground-state uniqueness or determine the nature of the excitation spectrum.